Viscosity and Carbon Black Level

How the addition of small solid particles to a liquid affects its viscosity (and other properties) is a long lasting problem, with many theoretical attempts. With respect to the above sections, that underlined specific aspects due to CB nature and characteristics, it is clear that no global understanding can yet be expected, that would be valid whatever the fluid and the filler. Let us consider that the fluid is a polymer (melt) then the complexity of the problem can nevertheless be stated in the form of a general functional, as follows  

As a first approach, one may fix parameters S, X, and T. In other terms, one considers how the viscosity of a Newtonian fluid, in isothermal conditions, is affected by the volume fraction of equal diameter spherical particles. With Tlpoiym and Hcpd the viscosities respectively of the pure polymer and of the fill compoimd, the following quantities are defined  

With reinforcing carbon blacks, the best spatial envelope for an aggregate is the ellipsoid and, owing to its mathematical simplicity, models proposed by Guth and Gold for spheres and rigid revolution ellipsoids drew the attention of several authors. For instance. White and Crowder considered the following equation  

Effect of Fillers on (Newtonian) Viscosity—Theoretical Approaches 

Rigid spheres with equal r gp=2.5 I (Einstein) diameter 

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